Geodesic
Solution to a~stochastic Darcy equation by the polynomial chaos expansion
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 20 (2017) no. 3, pp. 313-327

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This paper deals with the solution of a boundary value problem for the Darcy equation with a random hydraulic conductivity field. We use an approach based on the polynomial chaos expansion in the probability space of input data. We use the probabilistic collocation method to calculate the coefficients of the polynomial chaos expansion. A computational complexity of this algorithm is defined by the order of a polynomial chaos expansion and the number of terms in the Karhunen–Loève expansion. We calculate different Eulerian and Lagrangian statistical characteristics of the flow by the Monte Carlo and probabilistic collocation methods. Our calculations show a significant advantage of the probabilistic collocation method in comparison with the conventional direct Monte Carlo algorithm. 
@article{SJVM_2017_20_3_a6,
     author = {I. A. Shalimova and K. K. Sabelfeld},
     title = {Solution to a~stochastic {Darcy} equation by the polynomial chaos expansion},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {313--327},
     publisher = {mathdoc},
     volume = {20},
     number = {3},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2017_20_3_a6/}
}
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I. A. Shalimova; K. K. Sabelfeld. Solution to a~stochastic Darcy equation by the polynomial chaos expansion. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 20 (2017) no. 3, pp. 313-327. http://geodesic.mathdoc.fr/item/SJVM_2017_20_3_a6/